2022-03-29

## Catalan symmetric functions

Catalan symmetric functions were introduced in [Che10] and [Pan10]. These functions are $GL_\ell$-equivariant Euler characteristics of vector bundles on the flag variety. The Catalan symmetric functions specialize to $k$-Schur functions, see [BMPS18].

### Definition

Let $R_{ij}$ be the raising operators on Schur functions, so that

\[ R_{ij} \schurS_{\alpha} = \schurS_{\alpha+\varepsilon_i-\varepsilon_j}. \]Here, $\alpha$ can be a composition, and we then evaluate the Schur function using the Jacobi–Trudi formula.

A root ideal $\Phi$ is a an upper order ideal in

\[ \Delta_\ell^+ \coloneqq \{ (i,j) : 1 \leq i \lt j \leq \ell \} \]with the partial order relation $(a,b) \leq (c,d)$ when $a\geq c$ and $b \leq d.$ There are $\catalan(\ell)$ different such order ideals, thus explaining the name.

**Example.**

For example $\{ 15, 25, 35, 14, 24 \}$ is an order ideal, as they are entries above a Dyck path in the diagram:

$ \mathbf{15}$ | $ \mathbf{25}$ | $ \mathbf{35}$ | $45$ | $5$ |

$ \mathbf{14}$ | $ \mathbf{24}$ | $34$ | $4$ | |

$13$ | $23$ | $3$ | ||

$12$ | $2$ | |||

$1$ |

See also how area sequences of length $\ell$ are related to unit interval graphs.

Let $\gamma \in \setZ^\ell$ and let $\Phi$ be a root ideal. The Catalan symmetric function is then defined as

\[ \catalanH_{\Phi,\gamma}(\xvec;t) \coloneqq \prod_{(i,j)\in \Phi} (1-t R_{ij})^{-1}\schurS_{\gamma}(\xvec). \]Note that

\[ (1-rR_{ij})^{-1} = 1 + tR_{ij} + t^2 (R_{ij})^2 + t^3 (R_{ij})^3 + \dotsb \]but one only has to apply a finite number of these as $(R_{ij})^k$ kills any Schur function for sufficiently large $k.$ By construction, $\catalanH_{\Phi,\gamma}(\xvec;t)$ is a symmetric function.

The Catalan symmetric functions generalize the transformed Hall–Littlewood polynomials. We have that if $\mu$ is a partition of $n,$ then we use the full set of roots and have

\[ \hallLittlewoodT_{\mu}(\xvec;q) = \catalanH_{\Delta_n^+,\mu}(\xvec;q). \]**Example.**

For example, $\Phi = \{ 15, 25, 35, 14, 24 \}$ and $\gamma=(4,2,1,1,0)$ gives

\[ \catalanH_{\Phi,\gamma}(\xvec;t) = \schurS_{4211} + t \schurS_{431} + t \schurS_{521} \]In [Che10], it was conjectured that $\catalanH_{\Phi,\mu}(\xvec;t)$ is Schur-positive for any $\Phi$ and partition $\mu.$ This conjecture is resolved by J. Blasiak, J. Morse and A. Pun in [BMP20]. They introduce a larger family of non-symmetric Catalan functions, the tame non-symmetric Catalan functions, $\catalanH_{\Phi,\mu,w}(\xvec;t),$ which depend on an additional parameter $w \in \symS_n.$ It is then proved that $\catalanH_{\Phi,\mu,w}(\xvec;t)$ are key positive, which then implies the Schur positivity for $\catalanH_{\Phi,\mu}(\xvec;t).$

## Katalan symmetric functions

In [BMS20], the authors consider a $K$-theoretic version of the Catalan symmetric functions, named Katalan symmetric functions and show that this family includes the $K$-$k$-Schur functions, and the usual Catalan symmetric functions.

A nice conjecture regarding the Katalan symmetric functions is solved here: https://arxiv.org/pdf/2203.14483.pdf.

## $k$-Schur polynomials

The $k$-Schur functions were introduced in [LLM03] (under a different name and the notation $A_\mu^{(k)}[\xvec;t]$ was used). The motivation was to provide a strong refinement of the Schur positivity conjecture of the modified Macdonald polynomials. Several alternative definitions of $k$-Schur functions has since then surfaced, and not all have been proved to be equivalent. For a thorough reference, see the book [LLMSSZ14].

For each integer $k,$ we have the family of $k$-Schur functions $\{ \kSchur^{(k)}_\lambda(\xvec) \}$ where the $\lambda$ are $k$-bounded partitions, meaning that $\lambda_1 \leq k.$ The $k$-Schur functions form a basis in the subring of symmetric functions, spanned by $\completeH_1,\dotsc,\completeH_k,$ the complete homogeneous symmetric functions.

In [LM07], it is shown that whenever the hook-length of $\lambda$ is no larger than $k,$ we have the identity

\[ \kSchur^{(k)}_\lambda(\xvec) = \schurS_\lambda(\xvec). \]Hence, as $k\to \infty,$ the $k$-Schur functions reduce to the usual Schur functions.

### Definition

Note that there are several different, but conjectured equivalent definitions of $k$-Schur functions. We use the definition in [BMPS18], which defines the $k$-Schur functions as Catalan symmetric functions for special root ideals. Let $\mu$ be a partition with at most $\ell$ parts and $\mu_1 \leq k.$

Let $\Phi_\mu \coloneqq \{ (i,j) \in \Delta_\ell^+ : k-\mu_i+i \lt j \}$ and define the $k$-Schur functions as

\[ \kSchur^{(k)}_\mu(\xvec;t) \coloneqq \catalanH_{\Phi_\mu,\mu}(\xvec;t) = \prod_{i=1}^\ell \prod_{j=k+1-\mu_i+i}^\ell (1-tR_{ij})^{-1} \schurS_\mu. \]The specialization $t=1$ are also called $k$-Schur functions,

\[ \kSchur^{(k)}_\mu(\xvec) \coloneqq \kSchur^{(k)}_\mu(\xvec;1). \]Most results so far concern this specialization.

**Example.**

We have the following Schur expansions of some $k$-Schur functions:

\[ \kSchur^{(2)}_{221}(\xvec) = \schurS_{221}(\xvec)+\schurS_{311}(\xvec)+2\schurS_{320}(\xvec)+2\schurS_{410}(\xvec)+\schurS_{500}(\xvec) \] \[ \kSchur^{(3)}_{221}(\xvec) = \schurS_{221}(\xvec)+\schurS_{320}(\xvec)\quad \text{ and } \quad \kSchur^{(4)}_{221}(\xvec) = \schurS_{221}(\xvec) \]### Relation with affine Stanley symmetric functions

In [Lam06a] it is shown that the $k$-Schur functions are dual to the affine Schur functions.

### Pieri rule

L. Lapointe and J. Morse proved the following Pieri rule.

**Theorem (See [Thm. 29, LM07]).**

Let $\nu$ be a $k$-bounded partition and $r \leq k.$ Then

\[ \completeH_r \kSchur^{(k)}_\nu = \sum_{\mu \in H^k_{\nu,r}} \kSchur^{(k)}_\mu \]where $H^k_{\nu,r}$ is a certain subset of partitions formed by adding horizontal $r$-strips to $\lambda.$ To be more precise

\[ H^k_{\nu,r} = \{ \mu : \mu/\nu \text{ is a horizontal strip and } \mu^{(k)}/\nu^{(k)} \text{ is a vertical $r$-strip} \}. \]Here, $\mu^{(k)}$ denotes the $k$-conjugate of $\mu.$

### Murnaghan–Nakayama rule

J. Bandlow, A. Schilling and M. Zabrocki prove the following analogue of the Murnaghan–Nakayama rule.

**Theorem (See [BSZ11]).**

For $1\leq r \leq k$ and $\lambda$ being a $k$-bounded partition,

\[ \powerSum_r \kSchur^{(k)}_\lambda = \sum_{\mu} (-1)^{\ht(\mu/\lambda)} \kSchur^{(k)}_\mu \]where the sum is over all $k$-bounded partitions $\mu$ such that $\mu/\lambda$ is a $k$-ribbon of size $r.$ The definition of a $k$-ribbon is somewhat involved, see [BSZ11] for details.

An alternative proof is given by Lee in [Thm. 10.1, Lee15].

### Littlewood–Richardson rule

It is conjectured that the coefficients $c^{\nu,k}_{\lambda\mu}$ in

\[ \kSchur^{(k)}_\lambda \kSchur^{(k)}_\mu = \sum_{\nu : \nu_1 \leq k} c^{\nu,k}_{\lambda\mu} \kSchur^{(k)}_\nu \]are all non-negative. These coefficients 3-point Gromow–Witten invariants, see [LM08], and thus sometimes proved to be non-negative.

### Schur expansion

**Conjecture (See [LM07]).**

It is conjectured that

\[ \kSchur^{(k)}_\lambda(\xvec) = \schurS_\lambda + \sum_{\mu \triangleright \lambda } d^{k}_{\lambda\mu} \schurS_{\mu}(\xvec) \]with non-negative $d^{k}_{\lambda\mu}.$

This conjecture was recently proved, see this preprint.

## References

- [BMP20] Jonah Blasiak, Jennifer Morse and Anna Pun. Demazure crystals and the Schur positivity of Catalan functions. arXiv e-prints, 2020.
- [BMPS18] Jonah Blasiak, Jennifer Morse, Anna Pun and Daniel Summers. $k$-Schur expansions of Catalan functions. arXiv e-prints, 2018.
- [BMS20] Jonah Blasiak, Jennifer Morse and George H. Seelinger. $k$-theoretic Catalan functions. arXiv e-prints, 2020.
- [BSZ11] Jason Bandlow, Anne Schilling and Mike Zabrocki. The Murnaghan–Nakayama rule for k-Schur functions. Journal of Combinatorial Theory, Series A, 118(5):1588–1607, July 2011.
- [Che10] Li-Chung Chen. Skew-linked partitions and a representation-theoretic model for $k$-Schur functions. UC Berkeley. 2010.
- [Lam06a] Thomas Lam. Affine Stanley symmetric functions. American Journal of Mathematics, 128(6):1553–1586, 2006.
- [Lee15] Seung Jin Lee. Combinatorial description of the cohomology of the affine flag variety. arXiv e-prints, 2015.
- [LLM03] Luc Lapointe, Alain Lascoux and Jennifer Morse. Tableau atoms and a new Macdonald positivity conjecture. Duke Mathematical Journal, 116(1):103–146, January 2003.
- [LLMSSZ14] Thomas Lam, Luc Lapointe, Jennifer Morse, Anne Schilling, Mark Shimozono and Mike Zabrocki. k-Schur functions and affine Schubert calculus (fields institute monographs). Springer, 2014.
- [LM07] Luc Lapointe and Jennifer Morse. A k-tableau characterization of k-Schur functions. Advances in Mathematics, 213(1):183–204, August 2007.
- [LM08] Luc Lapointe and Jennifer Morse. Quantum cohomology and the k-Schur basis. Transactions of the American Mathematical Society, 360(4):2021–2040, 2008.
- [Pan10] Dmitri I. Panyushev. Generalised Kostka–Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles. Selecta Mathematica, 16(2):315–342, April 2010.